Note on the space part of anti-symmetric wave functions in the many-electron problem

It is pointed out that if a many-electron antisymmetric wave function is expanded as a sum of spin-product functions, each multiplied by a function of coordinates, the resulting functions of coordinates have many of the same useful features found with the symmetric and antisymmetric functions representing singlet and triplet states in a two-electron system. For finding the energy, or any function of coordinates only, in the approximation in which spin-orbit interaction is neglected, one such function of coordinates can be used, the spins being disregarded. Simple procedures allow one to find matrix components of such operators as S ² and L . S from the functions of coordinates. These procedures are much easier to visualize than the use of projection operators, the permutation group, or other methods in current use. The general procedures are illustrated by application to the three-electron problem of the lithium atom, as treated by Lunell, Kaldor, and Harris, and their application to the contact hyperfine structure is pointed out.     

Bond length alternation in cyclic polyenes. II. Unrestricted hartree–fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian, together with an empirical quasi harmonic σ-core potential is investigated using the unrestricted Hartree–Fock wave function employing different spatial orbitals for different spins. It is shown that in contrast to the restricted Hartree–Fock method, which favors bond alternation in large cyclic polyenes, the unrestricted Hartree–Fock method stabilizes the symmetric structures with equidistant internuclear separation. An assessment of the amount of correlation error recovered by the unrestricted Hartree–Fock procedure is examined and the qualitatively different behavior of the cyclic polyene models when described by restricted and unrestricted Hartree–Fock wave functions is discussed from this viewpoint.     

Unrestricted-Hartree–Fock wave functions approximating low-lying covalent states of ring π systems

We show that spin projected unrestricted-Hartree–Fock (PUHF ) wave functions are able to well approximate some low-lying covalent states of ring π systems. The UHF wave functions belong to either the axial or torsional spin density wave class. Their spin structures are found to be approximations for the spin correlation structures of the corresponding exact wave functions. The PUHF wave functions become close to exact eigenstates with homopolar valence bond characters in the strong correlation limit.     

Relation of the charge and spin correlation functions to the structures of π electron wave functions in alternant hydrocarbons

We analyze, in the Pariser–Parr–Pople (PPP ) model of alternant hydrocarbons, how the charge and spin correlation functions (CF 's) are related to the structure of a CI wave function and the MO 's of the systems. The analysis is based on the fact that an uncorrelated electron present in an orbital does not contribute to the linked dynamically correlated parts of the CF 's. By using the fact, simple rules are deduced predicting the covalent or ionic nature of the correlation structure in a low-lying state with two correlated electrons. The rules predict that the singlet minus and triplet plus states are covalent, while singlet plus and triplet minus states are ionic, where the plus and minus mean the alternancy symmetry. The rules also give a prediction for the unknown charge and spin correlation structures between different sites.     

Charge and spin correlation structures of conjugated π systems: Analysis of full CI wave functions of the PPP hamiltonian

An analysis of the electronic correlation structures by means of the charge and spin correlation functions is carried out for full CI wave functions of four, five, and six membered conjugated π systems described by the Pariser–Parr–Pople Hamiltonian. The low-lying states of these systems are classified as covalent (CV ) and ionic (IN ) states depending on whether the probability of finding two electrons simultaneously at the same position is small or large. It is found that many of excited CV states, the typical ones of which are the 2¹A_g state of linear π systems, have stronger CV character than the ground CV state, and their spin coupling structures are different from each other as well as from that of the ground CV state. The spin coupling structure in the ground CV state has an “antiferromagnetic” spin arrangement in favor of antiparallel coupling between nearest neighbor spins while in excited CV states the extent of the antiparallel spin coupling between nearest neighbor sites is decreased. IN states, which are less common for low-lying states than CV ones, are also found to have characteristic modulations in the charge correlation. In particular, the charge correlations in the lowest singlet IN states, 1¹B_u of linear π systems, 1¹B_2g of cyclobutadiene and 1¹B_1U of benzene, are alternating.     

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.     

The distribution of electronic charge in the ground state of cubic boron nitride

The distribution of electronic charge in cubic boron nitride is investigated using the bond orbital wave functions recently calculated by Coulson and Doggett. Plots of the one-electron density function, in the (110) plane, are found to be insensitive to the choice of atomic basis functions, in contradistinction to the previously calculated effective atomic charges. A number of structure amplitudes are also calculated for each of the bond orbital wave functions.     

Two-dimensional Raman spectra of atomic solids and liquids

We calculate third- and fifth-order Raman spectra of simple atoms interacting through a soft-core potential by means of molecular-dynamics (MD) simulations. The total polarizability of molecules is treated by the dipole-induced dipole model. Two- and three-body correlation functions of the polarizability at various temperatures are evaluated from equilibrium MD simulations based on a stability matrix formulation. To analyze the processes involved in the spectroscopic measurements, we divide the fifth-order response functions into symmetric and antisymmetric integrated response functions; the symmetric one is written as a simple three-body correlation function, while the antisymmetric one depends on a stability matrix. This analysis leads to a better understanding of the time scales and molecular motions that govern the two-dimensional (2D) signal. The 2D Raman spectra show novel differences between the solid and liquid phases, which are associated with the decay rates of coherent motions. On the other hand, these differences are not observed in the linear Raman spectra.     

Factored wave function for bound S-type states of two-electron atomic systems

A simple and accurate variational wave function in which the dependence in the interelectronic distance is factored is proposed to describe S-type states of two-electron atomic systems. We introduce a parameterization which generalizes the previous ones used in this same framework and which allows us to obtain in a simple way the wave function of both symmetric and antisymmetric excited states. We performed a systematic analysis of some exact properties such as the virial theorem and the cusp conditions and a study of both the one- and two-body densities. Finally, a comparison among the different correlation functions for these states was performed for helium. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 405–413, 1998     

Geminal model chemistry. IV. Variational and size consistent pure spin states

We present a computationally inexpensive method that yields ground state wave functions of pure spin symmetry. The method is variational and rigorously size consistent, free from adjustable parameters, and has a favorable scaling with system size. It is based on the recently introduced partially spin restricted geminal wave functions with limited spin contamination. Computations of a bond breaking, a transition metal compound, and a symmetric hydrogen cluster confirm the properties of this method.     

Computer simulation of quantum dynamics in a classical spin environment

In this paper, a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum–classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author’s opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nanoscience.     

Spin-coupled theory for 'N electrons in M orbitals' active spaces

Spin-coupled (SC) theory, an ab initio valence bond (VB) approach which uses a compact and an easy-to-interpret single-orbital product wave function comparable in quality to a ‘N in N’ complete-active-space self-consistent field [CASSCF(N,N)] construction, is extended to ‘N in M’ (N ≠ M) active spaces. The SC(N,M) wave function retains the essential features of the original SC model: It involves just the products of nonorthogonal orbitals covering all distributions of N electrons between M orbitals in which as few orbitals as possible, |N – M|, are doubly occupied (for N > M) or missing (for N < M) and all other orbitals are singly occupied; each of these products is combined with a flexible spin function which allows any mode of coupling of the spins of the orbitals within the product. The SC(N,M) wave function remains much more compact than a CASSCF(N,M) construction; for example, the SC(6,7) wave function includes 35 configuration state functions (CSFs) as opposed to the 490 CSFs in the CASSCF case. The essential features of the SC(N,M) method are illustrated through a SC(6,5) calculation on the cyclopentadienyl anion, C5H5(–), and a SC(6,7) calculation on the tropylium cation, C7H7(+). The SC(6,5) and SC(6,7) wave functions for C5H5(–) and C7H7(+) are shown to provide remarkably clear modern VB models for the electronic structures of these aromatic cyclic ions which closely resemble the well-known SC model of benzene and yet recover almost all of the correlation energy included in the corresponding CASSCF(6,5) and CASSCF(6,7) wave functions: over 97% in the case of C5H5(–) and over 95% in the case of C7H7(+).     

Dynamical hydrogen atom tunneling in dichlorotropolone: a combined quantum, semiclassical, and classical study

Based on the Cartesian Reaction Surface framework we construct a four-dimensional potential for the tropolone derivative 3,7-dichlorotropolone, a molecule with an intramolecular O-H...O hydrogen bond. The reduced configuration space involves the in-plane hydrogen atom coordinates, a symmetric O-O vibrational mode, and an antisymmetric mode related to deformations of the seven-membered ring. The system is characterized in terms of quantum mechanical computations of the low-lying eigenstates as well as a classical and semiclassical analysis of spectra obtained via Fourier transforming autocorrelation functions. For the semiclassical analysis we utilize the amplitude-free correlation function method [K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003)]. Our results demonstrate substantial anharmonic couplings leading to highly correlated wave functions even at moderate energies. Furthermore, the importance of dynamical tunneling in tropolone is suggested since many low-lying states--including the ground state--lie above the classical saddle point but nevertheless appear as split pairs.     

Configuration interaction matrix elements. I. Algebraic approach to the relationship between unitary group generators and permutations

Matrix elements of unitary group generators between spin-adapted antisymmetric states are shown to be proportional to spin matrix elements of so-called “line-up” permutations. The proportionality factor is given explicitly as a simple function of the orbital occupation numbers. If one bases the theory on ordered orbital products, the line-up permutations are given a priori. The final formulas have a very simple structure; this is a direct consequence of the fact that the spin functions have been taken to be geminally antisymmetric.     

Employing broken symmetry effects from unrestricted coupled cluster wave function to determine dynamic and non-dynamic electron correlation during triple bond breaking in the N2 molecule

The cluster structure of the singlet full symmetric component of the unrestricted Hartree-Fock (UHF)-based CCSD wave function describing the triple bond breaking in the nitrogen molecule has been subjected to a detailed analysis for the possibility of determining the dynamic and near-degeneracy electron correlation. The results obtained show that the spin and symmetry contaminations are not responsible only for the appearance of the artificial hump in the potential energy curve (PEC) generated by the UHF-based CCSD calculations. A theoretical analysis of this issue indicates that the UHF-based CCSD wave function and its singlet full symmetric component have a multi-reference structure. This form of the wave function allows to explain the mechanism for creating cluster contributions in the projected UHF-based CCSD wave function, which also provides the opportunity to explain the cause of the hump in the nitrogen PEC generated by the UCCSDecCCSD method.     

Curvilinear and surficial electron holes in atoms and molecules

The antisymmetric property of many-electron wave functions results in the well-known Fermi hole, which implies that any two electrons with the same spin cannot be at the same point in space. We here point out that for certain types of antisymmetric wave functions, there exist curvilinear and surficial electron holes which imply that two electrons cannot be on particular curves and surfaces in space.     

Rotation of methyl radicals in a solid argon matrix

Electron spin resonance (ESR) measurements were carried out to study the rotation of methyl radicals (CH3) in a solid argon matrix at 14-35 K temperatures. The radicals were produced by dissociating methane by plasma bursts generated either by a focused 193 nm laser radiation or a radio frequency discharge device during the gas condensation on the substrate. The ESR spectrum exhibits axial symmetry at the lowest temperature and is ascribed to ground state molecules with symmetric total nuclear spin function I=3/2. The hyperfine anisotropy (Aparallel)-Aperpendicular) was found to be -0.01 mT, whereas that of the g value was 2.5x10(-5). The anisotropy is observed for the first time in Ar and is manifested by the splitting of the low-field transition. Elevation of temperature leads reversibly to the appearance of excited state contribution having antisymmetric I=1/2. As a function of the sample temperature, the relative intensities of symmetric and antisymmetric spin states corresponding to ground and excited rotor states, respectively, proton hyperfine and electron g-tensor components, and spin-lattice relaxation rates were determined by a numerical fitting procedure. The experimental observations were interpreted in terms of a free rotation about the C3 axis and a thermal activation of the C2-type rotations above 15 K. The ground and excited rotational state energy levels were found to be separated by 11.2 cm-1 and to exhibit significantly different spin-lattice coupling. A crystal field model has been applied to evaluate the energy levels of the hindered rotor in the matrix, and crystal field parameter varepsilon4=-200 cm-1, corresponding to a 60 cm-1 effective potential barrier for rotation of the C3 axis, was obtained.     

The creation of off-diagonal elements in chemically induced dynamic nuclear polarization experiments

The nuclear density matrix created in pulsed CIDNP experiments contains off-diagonal elements whenever the resulting nuclear spin system is strongly coupled. These off-diagonal elements, which connect spin states with the same magnetic quantum number, are due to the mixing of nuclear state functions during the process of product formation. The observation of these elements by means of a two-pulse experiment is described.